Friday, February 24, 2017

Knowledge: Approximate or Exact?

Over at Greenbaggins, a discussion has arisen concerning the nature of our knowledge. Lane Keister's post seeks to contribute to the Clark-van Til debate. Lane suggests that God's knowledge forms a coherent whole that forms a context for all facts that He knows. Hence, our knowledge (lacking said context) cannot be identical to His, even in the knowledge of a single fact. Historically, the fact that van Til and Clark disagreed over was "2 + 2 = 4."

 This led to a discussion in the combox as to the putative differences between God's knowledge and ours over brute facts, such as

  "Jacques" is a longer word than "Jeff."

 I have suggested that there are four major differences.

 (1) God's knowledge is infallible, or non-contingent. Ours is contingent, hence fallible. 

 (2) God's knowledge is infinitely precise; ours is not. 

 (3) God's knowledge has complete context; ours does not (Lane's point). 

 (4) God's knowledge is immediate, with no need to reason or discover; ours is not.

Three objections have arisen to the notion that our knowledge is imprecise, which is point (2) above.  I've justified the point thus: all of our knowledge is approximate because in every area of knowledge, we rely on words and symbols (which have a semantic range of meaning), sense experience (which is imprecise), and reasoning (which is subject to error). Hence, although we may know certain facts to a very high degree of certainty, we cannot claim exact knowledge of any fact. As a simple example, I observed that every single measurement we make is approximate. At one stroke, this makes all of our knowledge approximate in the sense mentioned above.

Here are the objections that have arisen.

  O1: "All (human) knowledge is approximate" is self-defeating

The first objection comes from Jacques, who observes that "the claim that our knowledge is an approximation is itself urged as exact knowledge." In other words, the concern is that "knowledge is approximate" is self-defeating. One thinks here of Plato's destruction of relativism ("'There is no absolute truth' is itself an absolute truth") or the more recent failure of empirical positivism ("Only statements that are empirically verifiable can be considered true" is not itself empirically verifiable). I've responded that it is acceptable for "knowledge is approximate" to be itself an approximate statement. However, the discussion has not blossomed beyond that point.

  O2: Obvious defeaters

 The second objection comes from Don and Ron, who observe that a fact such as "Jacques is a longer word than Jeff" can be resolved by simply counting, with no approximation required. This is a self-evident defeater to the claim that all knowledge is approximate. Likewise, Ron urges that although we may not be able to measure 12 inches to infinite precision, we can nevertheless know without approximation that a length is between 11 and 13 inches. Don wonders further in what way our knowledge of this fact could possibly be different from God's.

  O3: Meta-claims about confidence intervals

 Ron further objects that if we now speak of confidence intervals (for example, "this tile is 12 inches wide, plus or minus 0.1 inches"), then we have now created a new proposition that is known exactly.

I would like to open this space for discussion of the propositions (1) - (4) and the three objections. I believe they all have entirely adequate responses.

 In particular:

Response to O1

Suppose that "All knowledge is approximate" turns out to be true for only 99.9% of human knowledge. Perhaps there is some undeniable proposition (perhaps "I think, therefore I am"; or perhaps "'Jacques' is a longer string than 'Jeff'")) that is known absolutely and precisely. Perhaps there are even a large number of such propositions, comprising as large as 0.1% of human knowledge.

This outcome would not be disastrous for the claim "All knowledge is approximate." For given any particular proposition P, it would still be highly likely that P is approximate. And if non-approximate P's can be identified and distinguished from the approximate, then so much the better. That's the worst case.

In point of fact, it's not yet clear that any non-approximate P's have certainly been identified, for such a P would have to be

Criterion 1: Known without possibility of error (that is, an infinitesimally thin confidence interval), Criterion 2: Completely determined in unambiguous language, whether mathematical or otherwise (that is, infinitely precise in its boundary)

To my knowledge, no P meeting Criteria 1 and 2 has ever been produced.

In short: It is acceptable to say that "All knowledge is approximate" is itself theoretically approximate; and so far, no counterexamples have been produced.

Response to O2

Here, we need to dip into some statistics. Likely, Don and Ron will agree that counting letters is not itself an infallible process. If we gave out the string "Jacques" to 100,000 English-speaking adults and asked them how many letters are in that string, we are very likely to get at least a couple of responses of "6" or "8". Every process has an error rate, and many of those error rates have been studied by process engineers (eg here or here). For sake of example, let's suppose that we get back a response of 99,997 people who count 7 letters, two who count 6, and one who counts 8. I suspect this is generous. Then we can now ask the important question, "How many letters *are* in that string, anyway?" Based on the responses, we can calculate an average (6.99999) and a standard deviation (.00001). This gives us a delightfully narrow confidence interval: We can be 99.9% confident that the true number of letters lies between 6.99995 and 7.00003 letters -- in other words, 7. BUT If we are willing to consider extremely outside chances, there is a very, very small chance that our two people who counted 6 were correct, and everyone else was wrong. How small? Absurdly small -- my calculator gives a p value of "0", but it's actually about e^-350. Well, practically speaking, this is exact for all useful purposes. There is no meaningful risk that the two people who counted 6 are actually correct. "Jacques" has 7 letters, the end. And that's my larger point. The word "approximation" is not a synonym for "poor estimate." It means simply that there is some kind of error bar, no matter how small. And in this case, because the error bars are truly and absurdly small, Don and Ron understandably round down and simply say that "Jacques has seven letters, with no approximation." 

In reality, the approximation is so good that no one doubts it.

The point is that excellent approximations are sometimes confused for perfect precision, and I believe that O2 stems from that confusion.

Response to O3

Actually, the knowledge of the error bars is itself approximate. Interested readers can start here. There are two sources of uncertainty -- random error, and systematic. The random error is described in the formula in the link. The systematic error cannot be mathematically captured, and comes about from the possibility that our process is actually flawed in some way without our knowledge:

  • The ruler is inaccurately calibrated 
  • The user of the ruler is using the wrong procedure (This happens a lot in high school labs) 
  • The boundaries of the object to be measured are not precisely defined. 
So the statement "This tile is between 11 and 13 inches" is not itself exactly known; there is a chance (however small) that we are using the wrong ruler or the wrong method for reading that ruler.


Why does this matter to the theologian? It is certainly interesting philosophy, but the payoff for the theologian is to recognize that our readings of Scripture will always and only give fallible approximations to the original meaning. And it is for this reason that Confessions are needed and yet revisable. If we could perfectly and precisely know the meaning of Scripture, then solo scriptura would suffice, and Confessions would not be needed. If on the other hand, if a council could perfectly and precisely know the meaning of Scripture, then it would not be true that " All synods or councils, since the apostles' times, whether general or particular, may err; and many have erred. Therefore they are not to be made the rule of faith, or practice; but to be used as a help in both."

JRC

Wednesday, February 8, 2017

The OPC Report on Republication

In which David R and I (along with assorted kibitzers) discuss republication... The OPC Report is here. Read more...

Thursday, August 4, 2011

American Ladies

This year marked the second year with American Ladies Vanessa virginiensis. Our hostplants were more established, so I didn't find myself repeating last year's ritual of find Pearly Everlasting in the field every three days!

The new method for this year was to take small sections of Pussytoes Antennaria neglecta and root them in smaller pots. The former method was to simply add leaves to the container as needed; but this seemed to force the AL cats to make nests more frequently than they might do in nature. The new method allowed the cats to make nests in whole plants.


The method had mixed success. The major problem was that I inadvertently put a large cat in with several eggs and 1st instar cats. They all disappeared but one.

The overly hungry caterpillar...


The key to success seems to be to provide two plants per caterpillar.


Summer at the Butterfly House

So where have the postings gone?

Basically, we've been busy (gardening, life, work) or gone (PA, NH). But in between, we've had various butterfly visitors.

May saw the release of several Spicebush Swallowtail and Black Swallowtails that had overwintered as pupae.



The girls also tried their hand at raising Cabbage Whites (translation: daddy fed the caterpillars).

Then summer began in earnest. We had many, many American Lady eggs laid on Pussytoes. Some of those came inside and we experimented with using whole plants as food, with mixed success. A friend "donated" a clutch of (unwanted) Black Swallowtails on parsley. We successfully raised the caterpillars to chrysalises on a new hostplant, Golden Alexanders -- only to have all but two pupae die mysteriously. Variagated Fritillaries, Monarchs, Spicebush Swallowtails, and Orange Sulfurs were all raised.

The exciting moment of Summer 2011, however, was finding 25 Pipevine Swallowtail Battus philenor caterpillars on July 12 that had stripped some Virginia Snakeroot Aristolochea serpentaria bare. This was a great moment because our pipevines had gone without caterpillars for their entire four year lifespans.

The cats naturally came in. Here's one, looking rather alien-like:


They're even cuter en masse:


Leaving for vacation on July 20 required turning all of our caterpillars loose on various hostplants. One consolation was that there was yet another batch of Pipevine Swallowtail caterpillars on the large pipevine Aristolochea macrophylla. Sure enough, on return we found five of those caterpillars -- along with many chewed leaves and frass, which suggests that the original 25 did not all perish.

A significant new change was the creation of a large caterpillar cage -- THE Butterfly House -- that will keep caterpillars off the back porch. The five PVS cats are currently testing it out for me.


In the Yard 8/4/2011

Seen today between 1 and 3 PM (no pics):

5 Tiger Swallowtails Pterourus glaucus, including one dark-form female
1 Orange Sulfur Colias eurytheme
2 Cabbage Whites Pieris rapae
4 Buckeyes Junonia coenia
1 Variegated Fritillary Euptoieta claudia
3 Eastern Tailed Blues Cupido comyntas
1 Zabulon Skipper Poanes zabulon male
4 Tawny-Edged Skippers Polites themistocles, 3 male 1 female.
3 Peck's Skippers Polites peckius, all male.
1 Sachem Atalopedes campestris female.
several Silver-Spotted Skippers Epargyreus clarus

Oh: and one very large and fat Pipevine Swallowtail Battus philenor caterpillar. :)

JRC

Saturday, June 4, 2011

Hairstreak Season Begins

There was a fresh Banded Hairstreak Satyrium calanus in the yard today -- a dark form. This is much earlier than I usually see them. What does this portend for other hairstreak species this year?

JRC

Sunday, May 22, 2011

May 22, 2011

We went biking along the Potomac today. Wingstem Verbesina alternifola and Nettles Urtica sp. are in abundance there, along with Paw-paw trees. We saw several Eastern Commas, Spicebush Swallowtails, Zebra Swallowtails, Red Admirals, and a couple of Sulphurs. I looked on the Wingstem for caterpillars or nests of the Silvery Checkerspot Chlosyne nycteis, but no luck.

JRC